{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://tvm.d2l.ai/chapter_getting_started/install.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "import tvm\n",
    "from tvm import te\n",
    "import time\n",
    "import timeit\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from IPython import display\n",
    "try:\n",
    "  import mxnet as mx\n",
    "except:\n",
    "  pass\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "np.random.seed(0)\n",
    "n = 100\n",
    "a = np.random.normal(size=n).astype(np.float32)\n",
    "b = np.random.normal(size=n).astype(np.float32)\n",
    "c = a + b\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.2.2. Defining the TVM Computation\n",
    "Now let’s implement vector_add in TVM. The TVM implementation differs from above in two ways:\n",
    "\n",
    "We don’t need to write the complete function, but only to specify how each element of the output, i.e. c[i], is computed\n",
    "\n",
    "TVM is symbolic, we create symbolic variables by specifying their shapes, and define how the program will be computed\n",
    "\n",
    "In the following program, we first declare the placeholders A and B for both inputs by specifying their shapes, (n,), through tvm.te.placeholder. Both A and B are Tensor objects, which we can feed data later. We assign names to them so we can print an easy-to-read program later.\n",
    "\n",
    "Next we define how the output C is computed by tvm.compute. It accepts two arguments, the output shape, and a function to compute each element by giving its index. Since the output is a vector, its elements are indexed by integers. The lambda function defined in tvm.compute accepts a single argument i, and returns c[i], which is identical to c[i] = a[i] + b[i] defined in vector_add. One difference is that we don’t write the for-loop, which will be filled by TVM later.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tvm.te.tensor.Tensor, tvm.te.tensor.Tensor)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import tvm\n",
    "from tvm import te # te stands for tensor expression\n",
    "\n",
    "# Save to the d2ltvm package.\n",
    "def vector_add(n):\n",
    "    \"\"\"TVM expression for vector add\"\"\"\n",
    "    A = te.placeholder((n,), name='a')\n",
    "    B = te.placeholder((n,), name='b')\n",
    "    C = te.compute(A.shape, lambda i: A[i] + B[i], name='c')\n",
    "    return A, B, C\n",
    "\n",
    "A, B, C = vector_add(n)\n",
    "type(A), type(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.2.3. Creating a Schedule\n",
    "To run the computation, we need to specify how to execute the program, for example, the order to access data and how to do multi-threading parallelization. Such an execution plan is called a schedule. Since C is the output tensor, let’s create a default schedule on its operator and print the pseudo codes.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#[version = \"0.0.5\"]\n",
       "@main = primfn(a_1: handle, b_1: handle, c_1: handle) -> ()\n",
       "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
       "  buffers = {a: Buffer(a_2: Pointer(float32), float32, [100], []),\n",
       "             b: Buffer(b_2: Pointer(float32), float32, [100], []),\n",
       "             c: Buffer(c_2: Pointer(float32), float32, [100], [])}\n",
       "  buffer_map = {a_1: a, b_1: b, c_1: c} {\n",
       "  for (i: int32, 0, 100) {\n",
       "    c_3: Buffer(c_2, float32, [100], [])[i] = (a_3: Buffer(a_2, float32, [100], [])[i] + b_3: Buffer(b_2, float32, [100], [])[i])\n",
       "  }\n",
       "}\n",
       "\n",
       "#[metadata]\n",
       "{\n",
       "  \"root\": 1, \n",
       "  \"nodes\": [\n",
       "    {\n",
       "      \"type_key\": \"\"\n",
       "    }, \n",
       "    {\n",
       "      \"type_key\": \"Map\", \n",
       "      \"keys\": [\n",
       "        \"IntImm\"\n",
       "      ], \n",
       "      \"data\": [2]\n",
       "    }, \n",
       "    {\n",
       "      \"type_key\": \"Array\", \n",
       "      \"data\": [3, 4]\n",
       "    }, \n",
       "    {\n",
       "      \"type_key\": \"IntImm\", \n",
       "      \"attrs\": {\n",
       "        \"dtype\": \"bool\", \n",
       "        \"span\": \"0\", \n",
       "        \"value\": \"1\"\n",
       "      }\n",
       "    }, \n",
       "    {\n",
       "      \"type_key\": \"IntImm\", \n",
       "      \"attrs\": {\n",
       "        \"dtype\": \"bool\", \n",
       "        \"span\": \"0\", \n",
       "        \"value\": \"1\"\n",
       "      }\n",
       "    }\n",
       "  ], \n",
       "  \"b64ndarrays\": [], \n",
       "  \"attrs\": {\"tvm_version\": \"0.11.dev0\"}\n",
       "}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = te.create_schedule(C.op)\n",
    "tvm.lower(s, [A, B, C], simple_mode=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vector_add(a, b, c):\n",
    "    for i in range(n):\n",
    "        c[i] = a[i] + b[i]\n",
    "\n",
    "d = np.empty(shape=n, dtype=np.float32)\n",
    "vector_add(a, b, d)\n",
    "np.testing.assert_array_equal(c, d)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def get_abc(shape, constructor=None):\n",
    "    \"\"\"Return random a, b and empty c with the same shape.\n",
    "    \"\"\"\n",
    "    np.random.seed(0)\n",
    "    a = np.random.normal(size=shape).astype(np.float32)\n",
    "    b = np.random.normal(size=shape).astype(np.float32)\n",
    "    c = np.empty_like(a)\n",
    "    if constructor:\n",
    "        a, b, c = [constructor(x) for x in (a, b, c)]\n",
    "    return a, b, c\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = get_abc(100, tvm.nd.array)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod = tvm.build(s, [A, B, C])\n",
    "x = np.ones(2)\n",
    "y = tvm.nd.array(x)\n",
    "mod(a, b, c)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.3.1. Broadcast Add¶\n",
    "https://tvm.d2l.ai/chapter_common_operators/broadcast_add.html#ch-bcast-add"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A broadcast operator process two tensors in different shapes. Normally, one of the operands has a particular dimension to be 1, which will be broadcast along the corresponding dimension of the other operator to perform the given calculation. Common scalar calculations can all be broadcast, such as elementary arithmetic and logical operations. Fig. 3.1.1 illustrates one broadcast add case between two 2-dimensional tensors. Broadcast operators are commonly seen in deep learning workloads, e.g. batch normalization.\n",
    "\n",
    "# ![title](https://tvm.d2l.ai/_images/bcast_add.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Operator Optimizations on CPUs¶\n",
    "https://tvm.d2l.ai/chapter_cpu_schedules/index.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import tvm\n",
    "from tvm import te\n",
    "\n",
    "# Save to the d2ltvm package.\n",
    "def broadcast_add(shape1, shape2):\n",
    "    \"\"\"Broadcast add between two 2-dimensional tensors\n",
    "\n",
    "    shape1, shape2 : the shapes of the input tensors\n",
    "    \"\"\"\n",
    "    assert len(shape1) == 2 and len(shape2) == 2, \\\n",
    "        \"broadcast tensors should both be 2-dimension\"\n",
    "    for i in range(len(shape1)):\n",
    "        assert shape1[i] == shape2[i] or shape1[i] == 1 or shape2[i] == 1, \\\n",
    "            \"tensor shapes do not fit for broadcasting\"\n",
    "    A = te.placeholder(shape1, name='A')\n",
    "    B = te.placeholder(shape2, name='B')\n",
    "    m = shape1[0] if shape2[0] == 1 else shape2[0]\n",
    "    n = shape1[1] if shape2[1] == 1 else shape2[1]\n",
    "    f = lambda x, y: A[0 if shape1[0]==1 else x, 0 if shape1[1]==1 else y] + \\\n",
    "        B[0 if shape2[0]==1 else x, 0 if shape2[1]==1 else y]\n",
    "    C = te.compute((m, n), f, name='C')\n",
    "    return A, B, C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [3, 1], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [3, 4], [])}\n",
      "  buffer_map = {A_1: A, B_1: B} {\n",
      "  allocate(C: Pointer(global float32), float32, [12]), storage_scope = global;\n",
      "  for (x: int32, 0, 3) {\n",
      "    for (y: int32, 0, 4) {\n",
      "      let cse_var_1: int32 = ((x*4) + y)\n",
      "      C_1: Buffer(C, float32, [12], [], align=32)[cse_var_1] = (A_3: Buffer(A_2, float32, [3], [])[x] + B_3: Buffer(B_2, float32, [12], [])[cse_var_1])\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "m = 3\n",
    "n = 4\n",
    "shape1 = (m, 1)\n",
    "shape2 = (m, n)\n",
    "A, B, C = broadcast_add(shape1, shape2)\n",
    "s = te.create_schedule(C.op)\n",
    "print(tvm.lower(s, [A, B], simple_mode=True))\n",
    "mod = tvm.build(s, [A, B, C])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.2. Matrix Multiplication¶\n",
    "https://tvm.d2l.ai/chapter_common_operators/matmul.html\n",
    "![xxx](https://tvm.d2l.ai/_images/matmul_default.svg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1616997653618455e-05"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import timeit\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from IPython import display\n",
    "\n",
    "timer = timeit.Timer(setup='import numpy as np',\n",
    "                     stmt='np.zeros((4, 4))')\n",
    "timer.timeit(10)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import d2ltvm\n",
    "import numpy as np\n",
    "import tvm\n",
    "from tvm import te\n",
    "\n",
    "# Save to the d2ltvm package\n",
    "def matmul(n, m, l):\n",
    "    \"\"\"Return the computing expression of matrix multiplication\n",
    "    A : n x l matrix\n",
    "    B : l x m matrix\n",
    "    C : n x m matrix with C = A B\n",
    "    \"\"\"\n",
    "    k = te.reduce_axis((0, l), name='k')\n",
    "    A = te.placeholder((n, l), name='A')\n",
    "    B = te.placeholder((l, m), name='B')\n",
    "    C = te.compute((n, m),\n",
    "                    lambda x, y: te.sum(A[x, k] * B[k, y], axis=k),\n",
    "                    name='C')\n",
    "    return A, B, C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [100, 100], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [100, 100], [])}\n",
      "  buffer_map = {A_1: A, B_1: B} {\n",
      "  allocate(C: Pointer(global float32), float32, [10000]), storage_scope = global;\n",
      "  for (x: int32, 0, 100) {\n",
      "    for (y: int32, 0, 100) {\n",
      "      C_1: Buffer(C, float32, [10000], [])[((x*100) + y)] = 0f32\n",
      "      for (k: int32, 0, 100) {\n",
      "        let cse_var_2: int32 = (x*100)\n",
      "        let cse_var_1: int32 = (cse_var_2 + y)\n",
      "        C_1[cse_var_1] = (C_1[cse_var_1] + (A_3: Buffer(A_2, float32, [10000], [])[(cse_var_2 + k)]*B_3: Buffer(B_2, float32, [10000], [])[((k*100) + y)]))\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "n = 100\n",
    "A, B, C = matmul(n, n, n)\n",
    "s = te.create_schedule(C.op)\n",
    "print(tvm.lower(s, [A, B], simple_mode=True))\n",
    "mod = tvm.build(s, [A, B, C])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = d2ltvm.get_abc((100, 100), tvm.nd.array)\n",
    "mod(a, b, c)\n",
    "np.testing.assert_allclose(np.dot(a.asnumpy(), b.asnumpy()),\n",
    "                           c.asnumpy(), atol=1e-5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.3. Convolution¶\n",
    "https://tvm.d2l.ai/chapter_common_operators/conv.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.3.1. Padding\n",
    "As a prerequisite to convolution, let’s first implement padding, which visually surrounds the targeting tensor with a “shell” surrounding it. The padding values are normally 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def padding(X, ph, pw, val=0):\n",
    "    \"\"\"Pad X with the given value in 2-D\n",
    "\n",
    "    ph, pw : height and width padding\n",
    "    val : padding value, default 0\n",
    "    \"\"\"\n",
    "    assert len(X.shape) >= 2\n",
    "    nh, nw = X.shape[-2], X.shape[-1]\n",
    "    return te.compute(\n",
    "            (*X.shape[0:-2], nh+ph*2, nw+pw*2),\n",
    "            lambda *i: te.if_then_else(\n",
    "                te.any(i[-2]<ph, i[-2]>=nh+ph, i[-1]<pw, i[-1]>=nw+pw),\n",
    "                val, X[i[:-2]+(i[-2]-ph, i[-1]-pw)]),\n",
    "            name='PaddedX')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify the results for a 3-D tensor.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[[0. 0. 0. 0. 0. 0. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 0. 0. 0. 0. 0. 0.]]\n",
      "\n",
      " [[0. 0. 0. 0. 0. 0. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 1. 1. 1. 1. 0. 0.]\n",
      "  [0. 0. 0. 0. 0. 0. 0. 0.]]]\n"
     ]
    }
   ],
   "source": [
    "A = te.placeholder((2,3,4))\n",
    "B = padding(A, 1, 2)\n",
    "s = te.create_schedule(B.op)\n",
    "mod = tvm.build(s, [A, B])\n",
    "\n",
    "a = tvm.nd.array(np.ones((2,3,4), dtype='float32'))\n",
    "b = tvm.nd.array(np.empty((2,5,8), dtype='float32'))\n",
    "mod(a, b)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.3.2. Convolution\n",
    "\n",
    "![xxx](https://tvm.d2l.ai/_images/conv-stride.svg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def conv_out_size(n, k, p, s):\n",
    "    \"\"\"Compute the output size by given input size n (width or height),\n",
    "    kernel size k, padding p, and stride s\n",
    "    Return output size (width or height)\n",
    "    \"\"\"\n",
    "    return (n - k + 2 * p)//s + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def conv(oc, ic, nh, nw, kh, kw, ph=0, pw=0, sh=1, sw=1):\n",
    "    \"\"\"Convolution\n",
    "\n",
    "    oc, ic : output and input channels\n",
    "    nh, nw : input width and height\n",
    "    kh, kw : kernel width and height\n",
    "    ph, pw : height and width padding sizes, default 0\n",
    "    sh, sw : height and width strides, default 1\n",
    "    \"\"\"\n",
    "    # reduction axes\n",
    "    ric = te.reduce_axis((0, ic), name='ric')\n",
    "    rkh = te.reduce_axis((0, kh), name='rkh')\n",
    "    rkw = te.reduce_axis((0, kw), name='rkw')\n",
    "    # output height and width\n",
    "    oh = conv_out_size(nh, kh, ph, sh)\n",
    "    ow = conv_out_size(nw, kw, pw, sw)\n",
    "    # pad X and then compute Y\n",
    "    X = te.placeholder((ic, nh, nw), name='X')\n",
    "    K = te.placeholder((oc, ic, kh, kw), name='K')\n",
    "    PaddedX = padding(X, ph, pw) if ph * pw != 0 else X\n",
    "    Y = te.compute(\n",
    "        (oc, oh, ow),\n",
    "        lambda c, i, j: te.sum(\n",
    "            PaddedX[ric, i*sh+rkh, j*sw+rkw] * K[c, ric, rkh, rkw],\n",
    "            axis=[ric, rkh, rkw]), name='Y')\n",
    "    return X, K, Y, PaddedX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_conv_data(oc, ic, n, k, p=0, s=1, constructor=None):\n",
    "    \"\"\"Return random 3-D data tensor, 3-D kernel tenor and empty 3-D output\n",
    "    tensor with the shapes specified by input arguments.\n",
    "\n",
    "    oc, ic : output and input channels\n",
    "    n : input width and height\n",
    "    k : kernel width and height\n",
    "    p : padding size, default 0\n",
    "    s : stride, default 1\n",
    "    constructor : user-defined tensor constructor\n",
    "    \"\"\"\n",
    "    np.random.seed(0)\n",
    "    data = np.random.normal(size=(ic, n, n)).astype('float32')\n",
    "    weight = np.random.normal(size=(oc, ic, k, k)).astype('float32')\n",
    "    on = conv_out_size(n, k, p, s)\n",
    "    out = np.empty((oc, on, on), dtype='float32')\n",
    "    if constructor:\n",
    "        data, weight, out = (constructor(x) for x in [data, weight, out])\n",
    "    return data, weight, out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(X_1: handle, K_1: handle, Y_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {X: Buffer(X_2: Pointer(float32), float32, [6, 12, 12], []),\n",
      "             K: Buffer(K_2: Pointer(float32), float32, [4, 6, 3, 3], []),\n",
      "             Y: Buffer(Y_2: Pointer(float32), float32, [4, 12, 12], [])}\n",
      "  buffer_map = {X_1: X, K_1: K, Y_1: Y} {\n",
      "  allocate(PaddedX: Pointer(global float32), float32, [1176]), storage_scope = global {\n",
      "    for (i0: int32, 0, 6) {\n",
      "      for (i1: int32, 0, 14) {\n",
      "        for (i2: int32, 0, 14) {\n",
      "          PaddedX_1: Buffer(PaddedX, float32, [1176], [])[(((i0*196) + (i1*14)) + i2)] = @tir.if_then_else(((((i1 < 1) || (i1 == 13)) || (i2 < 1)) || (i2 == 13)), 0f32, X_3: Buffer(X_2, float32, [864], [])[((((i0*144) + (i1*12)) + i2) - 13)], dtype=float32)\n",
      "        }\n",
      "      }\n",
      "    }\n",
      "    for (c: int32, 0, 4) {\n",
      "      for (i: int32, 0, 12) {\n",
      "        for (j: int32, 0, 12) {\n",
      "          Y_3: Buffer(Y_2, float32, [576], [])[(((c*144) + (i*12)) + j)] = 0f32\n",
      "          for (ric: int32, 0, 6) {\n",
      "            for (rkh: int32, 0, 3) {\n",
      "              for (rkw: int32, 0, 3) {\n",
      "                let cse_var_1: int32 = (((c*144) + (i*12)) + j)\n",
      "                Y_3[cse_var_1] = (Y_3[cse_var_1] + (PaddedX_1[(((((ric*196) + (i*14)) + (rkh*14)) + j) + rkw)]*K_3: Buffer(K_2, float32, [216], [])[((((c*54) + (ric*9)) + (rkh*3)) + rkw)]))\n",
      "              }\n",
      "            }\n",
      "          }\n",
      "        }\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "oc, ic, n, k, p, s = 4, 6, 12, 3, 1, 1\n",
    "X, K, Y, _ = conv(oc, ic, n, n, k, k, p, p, s, s)\n",
    "sch = te.create_schedule(Y.op)\n",
    "mod = tvm.build(sch, [X, K, Y])\n",
    "print(tvm.lower(sch, [X, K, Y], simple_mode=True))\n",
    "\n",
    "data, weight, out = get_conv_data(oc, ic, n, k, p, s, tvm.nd.array)\n",
    "mod(data, weight, out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time for np.zeros((4, 4)): 2.1449784109344806e-07\n"
     ]
    }
   ],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def bench_workload(workload):\n",
    "    \"\"\"Benchmark a workload\n",
    "\n",
    "    workload: a method that accept a num_repeat argument\n",
    "    and return its total execution time\n",
    "    \"\"\"\n",
    "    workload(1)  # warmup\n",
    "    time = workload(1)  # the time to run once\n",
    "    if time > 1: return time\n",
    "    # The number of repeats to measure at least 1 second\n",
    "    num_repeats = max(int(1.0 / time), 5)\n",
    "    return workload(num_repeats) / num_repeats\n",
    "\n",
    "print('time for np.zeros((4, 4)):', bench_workload(timer.timeit))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Vector Add¶\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.1. Setup\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from d2l_tvm import d2ltvm\n",
    "import inspect\n",
    "from IPython import display\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import timeit\n",
    "import tvm\n",
    "from tvm import te"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def plot(X, Y, xlabel=None, ylabel=None, legend=[], xlim=None,\n",
    "         ylim=None, xscale='linear', yscale='linear', fmts=None,\n",
    "         figsize=(4.5, 3)):\n",
    "    \"\"\"Plot multiple lines\"\"\"\n",
    "    display.set_matplotlib_formats('svg')\n",
    "    plt.rcParams['figure.figsize'] = figsize\n",
    "    axes = plt.gca()\n",
    "    X, Y = np.array(X), np.array(Y)\n",
    "    if X.shape != Y.shape: X = [X] * len(Y)\n",
    "    if not fmts: fmts = ['-'] * len(X)\n",
    "    for x, y, fmt in zip(X, Y, fmts):\n",
    "        axes.plot(x, y, fmt)\n",
    "    axes.set_xlabel(xlabel)\n",
    "    axes.set_ylabel(ylabel)\n",
    "    axes.set_xscale(xscale)\n",
    "    axes.set_yscale(yscale)\n",
    "    axes.set_xlim(xlim)\n",
    "    axes.set_ylim(ylim)\n",
    "    if legend: axes.legend(legend)\n",
    "    axes.grid()\n",
    "\n",
    "# Save to the d2ltvm package\n",
    "def plot_gflops(sizes, gflops, legend, xlabel='Size'):\n",
    "    d2ltvm.plot(sizes, gflops, xlabel=xlabel, ylabel='GFLOPS',\n",
    "             xscale='log', yscale='log',\n",
    "             legend=legend, fmts=['--']*(len(gflops)-1)+['-'])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
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   "source": [
    "sizes = 2**np.arange(5, 29, 4)\n",
    "np_add = lambda n: timeit.Timer(setup='import numpy as np\\n'\n",
    "                                'import d2ltvm\\n'\n",
    "                                'a, b, c = d2ltvm.get_abc(%d)' % n,\n",
    "                                stmt='np.add(a, b, out=c)')\n",
    "exe_times = [d2ltvm.bench_workload(np_add(n).timeit) for n in sizes]\n",
    "np_gflops = sizes / 1e9 / np.array(exe_times)\n",
    "plot_gflops(sizes, [np_gflops], ['numpy'])\n"
   ]
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    "3.2. Default Schedule\n",
    "In the following code block, we define a reusable method to benchmark TVM performance. It accepts three arguments: 1) a func which returns the schedule and its corresponding symbolic tensor arguments; 2) the size list specifying a number of the vector lengths; and 3) the machine target which is CPU-related for this chapter and will be GPU-related in the next chapter.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def bench_vector_add_tvm(func, sizes, target):\n",
    "    def workload(nrepeats):\n",
    "        timer = mod.time_evaluator(mod.entry_name, dev=ctx, number=nrepeats)\n",
    "        return timer(a, b, c).mean * nrepeats\n",
    "    times = []\n",
    "    for n in sizes:\n",
    "        s, (A, B, C) = func(int(n))\n",
    "        mod = tvm.build(s, [A, B, C], target)\n",
    "        ctx = tvm.device(target, 0)\n",
    "        a, b, c = d2ltvm.get_abc(n, lambda x: tvm.nd.array(x, device=ctx))\n",
    "        times.append(d2ltvm.bench_workload(workload))\n",
    "    return sizes / 1e9 / np.array(times)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(a_1: handle, b_1: handle, c_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {a: Buffer(a_2: Pointer(float32), float32, [64], []),\n",
      "             b: Buffer(b_2: Pointer(float32), float32, [64], []),\n",
      "             c: Buffer(c_2: Pointer(float32), float32, [64], [])}\n",
      "  buffer_map = {a_1: a, b_1: b, c_1: c} {\n",
      "  for (i: int32, 0, 64) {\n",
      "    c_3: Buffer(c_2, float32, [64], [])[i] = (a_3: Buffer(a_2, float32, [64], [])[i] + b_3: Buffer(b_2, float32, [64], [])[i])\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def default(n):\n",
    "    A, B, C = d2ltvm.vector_add(n)\n",
    "    s = te.create_schedule(C.op)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = default(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "; ModuleID = 'TVMMod'\n",
      "source_filename = \"TVMMod\"\n",
      "target datalayout = \"e-m:e-p270:32:32-p271:32:32-p272:64:64-i64:64-f80:128-n8:16:32:64-S128\"\n",
      "target triple = \"x86_64-unknown-linux-gnu\"\n",
      "\n",
      "%0 = type { i8*, %1, i32, %2, i64*, i64*, i64 }\n",
      "%1 = type { i32, i32 }\n",
      "%2 = type { i8, i8, i16 }\n",
      "\n",
      "@__TVMAPISetLastError = linkonce dllexport local_unnamed_addr global void (i8*)* null, align 8\n",
      "@.str = private constant [69 x i8] c\"Assert fail: (num_args == 3), default_function: num_args should be 3\\00\", align 1\n",
      "@.\n"
     ]
    }
   ],
   "source": [
    "target = 'llvm -mcpu=skylake-avx512'\n",
    "mod = tvm.build(s, args, target)\n",
    "print(mod.get_source()[:500])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
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   "source": [
    "default_gflops = bench_vector_add_tvm(default, sizes, target)\n",
    "plot_gflops(sizes, [np_gflops, default_gflops], ['numpy', 'default'])\n"
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    "When the vector size is small, the default scheduling outperforms NumPy, which means that in this method the function call overhead of TVM is smaller than NumPy. It’s not surprising to find the performance degrades when increasing the vector size as the data cannot fit into the last level cache.\n",
    "\n"
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    "3.3. Parallelization\n",
    "One important optimization that is not enabled by default is thread-level parallelization. The vector add operator is an embarrassingly parallel workload, we can just change the for-loop into a parallel for-loop. In TVM, we first obtain the scheduler for the output symbol C by s[C], and then impose the parallelization of the computation to its first axis, which is C.op.axis[0].\n",
    "\n"
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   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "@main = primfn(a_1: handle, b_1: handle, c_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {a: Buffer(a_2: Pointer(float32), float32, [64], []),\n",
      "             b: Buffer(b_2: Pointer(float32), float32, [64], []),\n",
      "             c: Buffer(c_2: Pointer(float32), float32, [64], [])}\n",
      "  buffer_map = {a_1: a, b_1: b, c_1: c} {\n",
      "  for (i: int32, 0, 64) \"parallel\" {\n",
      "    c_3: Buffer(c_2, float32, [64], [])[i] = (a_3: Buffer(a_2, float32, [64], [])[i] + b_3: Buffer(b_2, float32, [64], [])[i])\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def parallel(n):\n",
    "    s, (A, B, C) = default(n)\n",
    "    # add thread-level parallelism\n",
    "    s[C].parallel(C.op.axis[0])\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = parallel(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
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    "We can see that is changed to in the above pseudo codes. It means that the iterations could be executed in parallel by multiple threads. A typical implementation on a system with CPU cores is we first create threads with one thread for each core, then thread will execute blocks if . All these threads will run simultaneously on their exclusive cores to achieve parallelization, and come back to retrieve another block to execute after finishing one. This is often called the round-robin scheduling, which works well if each thread runs at the same speed and every iteration has a roughly same workload. \n",
    "\n",
    "TVM’s thread-level parallelization implementation is a special case of round-robin, which evenly divides the to-be-parallelized -length loop into blocks. Therefore, each thread only needs to process one block. Furthermore, the TVM runtime binds each working thread to a disjoint physical core to avoid resource contention and thread migration overhead. There are other thread-level parallelization schemes such as the more dynamic consumer-producer scheduling.forparallel "
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    "parallel_gflops = bench_vector_add_tvm(parallel, sizes, target)\n",
    "plot_gflops(sizes, [np_gflops, default_gflops, parallel_gflops],\n",
    "     ['numpy', 'default', 'parallel'])\n"
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    "Comparing the results we obtained before, parallelization significantly improves the performance when the workloads are large, e.g. vector lengths beyond . However, the parallelization overhead impact the performance for small workloads, where single thread is even faster. The performance drops at a larger size as multi-core comes in play, leading to a larger amount of L2 cache in total. "
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    "3.4. Vectorization\n",
    "A single core may have SIMD units to run multiple arithmetic operations at the same time as we saw in Section 1. Although one iteration in the above loop only has a single add operation, we can explicitly allocate more operations within an iteration, and ask the compiler to use SIMD instructions to process them.\n",
    "\n",
    "The way to do it is first splitting the one-level for-loop into a two-level nested for-loop using a . The inner loop consists of original iterations that will be grouped together accordingly to execute in SIMD instructions on a core. And iterations in the outer loop still run in parallel.factorfactor\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(a_1: handle, b_1: handle, c_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {a: Buffer(a_2: Pointer(float32), float32, [64], []),\n",
      "             b: Buffer(b_2: Pointer(float32), float32, [64], []),\n",
      "             c: Buffer(c_2: Pointer(float32), float32, [64], [])}\n",
      "  buffer_map = {a_1: a, b_1: b, c_1: c} {\n",
      "  for (i.outer: int32, 0, 8) \"parallel\" {\n",
      "    let cse_var_1: int32 = (i.outer*8)\n",
      "    c_3: Buffer(c_2, float32, [64], [])[ramp(cse_var_1, 1, 8)] = (a_3: Buffer(a_2, float32, [64], [])[ramp(cse_var_1, 1, 8)] + b_3: Buffer(b_2, float32, [64], [])[ramp(cse_var_1, 1, 8)])\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def vectorized(n):\n",
    "    s, (A, B, C) = default(n)\n",
    "    outer, inner = s[C].split(C.op.axis[0], factor=8)\n",
    "    s[C].parallel(outer)\n",
    "    s[C].vectorize(inner)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = vectorized(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the outer for-loop is reduced to 8 iterations, while the inner for-loop is vectorized by with a stride of 1 and width of 8. The definition of is inherited from Halide.rampramp\n",
    "\n",
    "Again, we check the performance of vectorization and plot the comparison diagram via the following code block.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
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   "source": [
    "vectorized_gflops = bench_vector_add_tvm(vectorized, sizes, target)\n",
    "plot_gflops(sizes, [np_gflops, default_gflops, parallel_gflops, vectorized_gflops],\n",
    "     ['numpy', 'default', 'parallel', 'vectorized'])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The performance of the vectorized version is almost as the plain parallelization version. It’s partially because the vector add is bottlenecked by memory bandwidth instead of computation, while SIMD only helps the latter. We will see it helps more on computation intensive workloads such as matrix multiplication later.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Broadcast Add¶\n",
    "https://tvm.d2l.ai/chapter_cpu_schedules/broadcast_add.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "%matplotlib inline\n",
    "import d2ltvm\n",
    "import inspect\n",
    "from IPython import display\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import timeit\n",
    "import tvm\n",
    "from tvm import te"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 450x300 with 1 Axes>"
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   "source": [
    "sizes = 2**np.arange(5, 15, 2)\n",
    "np_bcast_add = lambda s1, s2: timeit.Timer(setup='import numpy as np\\n'\n",
    "                                'import d2ltvm\\n'\n",
    "                                'a, b, c = d2ltvm.get_bcast_data(%s, %s)' % (s1, s2),\n",
    "                                stmt='np.add(a, b, out=c)')\n",
    "exe_times = [d2ltvm.bench_workload(np_bcast_add((n, 1), (n, n)).timeit) for n in sizes]\n",
    "np_gflops = sizes * sizes / 1e9 / np.array(exe_times)\n",
    "# data size in MB\n",
    "x_axis_sizes = (sizes * sizes * 2 + sizes * sizes) * 4 / 1e6\n",
    "d2ltvm.plot_gflops(x_axis_sizes, [np_gflops], ['numpy'], xlabel='Size (MB)')"
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   "metadata": {},
   "source": [
    "4.2. Good Schedule\n",
    "The following code block defines a benchmarking method for broadcast add of a specific pattern in TVM. It follows the same format we discussed in Section 3.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def bench_bcast_add_tvm(func, sizes, target):\n",
    "    def workload(nrepeats):\n",
    "        timer = mod.time_evaluator(mod.entry_name, dev=ctx, number=nrepeats)\n",
    "        return timer(a, b, c).mean * nrepeats\n",
    "    times = []\n",
    "    for n in sizes:\n",
    "        n = int(n)\n",
    "        s, (A, B, C) = func(n)\n",
    "        mod = tvm.build(s, [A, B, C], target)\n",
    "        ctx = tvm.device(target, 0)\n",
    "        a, b, c = d2ltvm.get_bcast_data((n, 1), (n, n), lambda x: tvm.nd.array(x, device=ctx))\n",
    "        times.append(d2ltvm.bench_workload(workload))\n",
    "    return sizes * sizes / 1e9 / np.array(times)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The good schedule (i.e. what works) follows the similar scheme defined in Section 3.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 1], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x: int32, 0, 64) \"parallel\" {\n",
      "    let cse_var_1: int32 = (x*64)\n",
      "    C_3: Buffer(C_2, float32, [4096], [])[ramp(cse_var_1, 1, 64)] = (broadcast(A_3: Buffer(A_2, float32, [64], [])[x], 64) + B_3: Buffer(B_2, float32, [4096], [])[ramp(cse_var_1, 1, 64)])\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "target = 'llvm -mcpu=skylake-avx512'\n",
    "def default(n):\n",
    "    A, B, C = d2ltvm.broadcast_add((n, 1), (n, n))\n",
    "    s = te.create_schedule(C.op)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "def good_schedule(n):\n",
    "    s, (A, B, C) = default(n)\n",
    "    x, y = C.op.axis\n",
    "    s[C].parallel(x)\n",
    "    s[C].vectorize(y)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = good_schedule(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
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   "source": [
    "good_gflops = bench_bcast_add_tvm(good_schedule, sizes, target)\n",
    "d2ltvm.plot_gflops(x_axis_sizes, [np_gflops, good_gflops], ['numpy', 'good'], xlabel='Size (MB)')\n"
   ]
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    "4.3. Bad Schedule\n",
    "\n",
    "Now let’s see how a bad schedule (i.e. what doesn’t work) looks like. Note that our data are all stored in row-major and our schedules always make sure that the innermost loop manipulates the data in consecutive space.\n",
    "\n",
    "However, if we think from the perspective vectorization, we may find that broadcasting A[x] to a vectorized register brings overhead. A better way is to have multiple values of A to add with multiple values of B in a vectorized instruction, which is illustrated in Fig. 4.3.1.\n",
    "\n",
    "![xxx](https://tvm.d2l.ai/_images/bcast_add_vectorize.svg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to make vectorization along the column, we need to reorder to data access pattern as follows.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 1], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (y: int32, 0, 64) \"parallel\" {\n",
      "    C_3: Buffer(C_2, float32, [4096], [])[ramp(y, 64, 64)] = (A_3: Buffer(A_2, float32, [64], [])[ramp(0, 1, 64)] + B_3: Buffer(B_2, float32, [4096], [])[ramp(y, 64, 64)])\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def bad_schedule(n):\n",
    "    s, (A, B, C) = default(n)\n",
    "    x, y = C.op.axis\n",
    "    s[C].reorder(y, x)\n",
    "    s[C].parallel(y)\n",
    "    s[C].vectorize(x)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = bad_schedule(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
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   "source": [
    "sizes = 2**np.arange(5, 11, 2)\n",
    "bad_gflops = bench_bcast_add_tvm(bad_schedule, sizes, target)\n",
    "diff = len(good_gflops)-len(bad_gflops)\n",
    "bad_gflops = np.append(bad_gflops, [np.nan] * diff)\n",
    "d2ltvm.plot_gflops(x_axis_sizes, [np_gflops, good_gflops, bad_gflops],\n",
    "     ['numpy', 'good', 'bad'],  xlabel='Size (MB)')\n"
   ]
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   "source": [
    "B的列数据不连续，带来搬运开销。\n",
    "Note that in order to make it finish in a short period of time, we reduce the number of tested sizes for this schedule.\n",
    "\n",
    "We can see that the performance of the vectorization-favorable schedule is pretty bad. The reason is that we access B in a stride to get the data in the same column, which is much slower than accessing data in a consecutive space. And the thread-level parallelism would make it even worse as different threads compete in getting the data in the same cache line.\n",
    "\n",
    "The result shows that a good schedule needs to consider multiple performance-related factors together. A scheduling scheme that favors one aspect may lead to bad overall performance as it harms other aspects.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. Matrix Multiplication¶\n",
    "We saw the NumPy dot operator nearly reaches the peak performance of our CPU (the Xeon E5-2686 v4) in Section 1. In this section, we will investigate multiple scheduling strategies for this operator in TVM.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import d2ltvm\n",
    "import numpy as np\n",
    "import timeit\n",
    "import tvm\n",
    "from tvm import te\n",
    "\n",
    "target = 'llvm -mcpu=skylake-avx512'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def np_matmul_timer(n):\n",
    "    timer = timeit.Timer(setup='import numpy as np\\n'\n",
    "                         'import d2ltvm\\n'\n",
    "                         'a, b, c = d2ltvm.get_abc(%s)' % str((n,n)),\n",
    "                         stmt = 'np.dot(a, b, out=c)')\n",
    "    return timer.timeit\n",
    "\n",
    "sizes = 2**np.arange(5, 12, 1)\n",
    "exe_times = [d2ltvm.bench_workload(np_matmul_timer(n)) for n in sizes]\n",
    "np_gflops = 2 * sizes **3 / 1e9 / np.array(exe_times)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5.2. Default Schedule\n",
    "The default schedule consists of three nested for-loops.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 64], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x: int32, 0, 64) {\n",
      "    for (y: int32, 0, 64) {\n",
      "      C_3: Buffer(C_2, float32, [4096], [])[((x*64) + y)] = 0f32\n",
      "      for (k: int32, 0, 64) {\n",
      "        let cse_var_2: int32 = (x*64)\n",
      "        let cse_var_1: int32 = (cse_var_2 + y)\n",
      "        C_3[cse_var_1] = (C_3[cse_var_1] + (A_3: Buffer(A_2, float32, [4096], [])[(cse_var_2 + k)]*B_3: Buffer(B_2, float32, [4096], [])[((k*64) + y)]))\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def default(n):\n",
    "    A, B, C = d2ltvm.matmul(n, n, n)\n",
    "    return te.create_schedule(C.op), (A, B, C)\n",
    "\n",
    "s, args = default(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save to the d2ltvm package.\n",
    "def bench_matmul_tvm(func, sizes, target):\n",
    "    def workload(nrepeats):\n",
    "        timer = mod.time_evaluator(mod.entry_name, dev=ctx, number=nrepeats)\n",
    "        return timer(a, b, c).mean * nrepeats\n",
    "    times = []\n",
    "    for n in sizes:\n",
    "        s, (A, B, C) = func(int(n))\n",
    "        mod = tvm.build(s, [A, B, C], target)\n",
    "        ctx = tvm.device(target, 0)\n",
    "        a, b, c = d2ltvm.get_abc((n, n), lambda x: tvm.nd.array(x, device=ctx))\n",
    "        times.append(d2ltvm.bench_workload(workload))\n",
    "    return 2 * sizes**3 / 1e9 / np.array(times)"
   ]
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   "execution_count": 43,
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   "source": [
    "default_gflops = bench_matmul_tvm(default, sizes, target)\n",
    "d2ltvm.plot_gflops(sizes, [np_gflops, default_gflops], ['numpy', 'default'])"
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   "source": [
    "5.3. Reordering Axes\n",
    "\n",
    "The first problem we can see from Fig. 3.2.1 is that matrix  B\n",
    "  is accessed column by column while its elements are stored by rows (i.e. matrix  B\n",
    "  is in row-major). In other words, in the pseudo code above, we iterate axis y before axis k. Simply switching these two for-loops will make all elements read and written sequentially. Fig. 5.3.1 illustrates the changed the data access pattern.\n",
    "\n",
    "![](https://tvm.d2l.ai/_images/matmul_reorder.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To implement it, we change the axes order from (x, y, k) to (x, k, y) by the reorder primitive. The corresponding pseudo code verifies that we are processing all matrices row by row now.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 64], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x: int32, 0, 64) {\n",
      "    for (y.init: int32, 0, 64) {\n",
      "      C_3: Buffer(C_2, float32, [4096], [])[((x*64) + y.init)] = 0f32\n",
      "    }\n",
      "    for (k: int32, 0, 64) {\n",
      "      for (y: int32, 0, 64) {\n",
      "        let cse_var_2: int32 = (x*64)\n",
      "        let cse_var_1: int32 = (cse_var_2 + y)\n",
      "        C_3[cse_var_1] = (C_3[cse_var_1] + (A_3: Buffer(A_2, float32, [4096], [])[(cse_var_2 + k)]*B_3: Buffer(B_2, float32, [4096], [])[((k*64) + y)]))\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def reorder(n):\n",
    "    s, (A, B, C) = default(n)\n",
    "    (x, y), (k,) = C.op.axis, C.op.reduce_axis\n",
    "    s[C].reorder(x, k, y)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = reorder(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the reordering significantly improves the performance compared to the default schedule.\n",
    "\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 45,
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   "source": [
    "reorder_gflops = bench_matmul_tvm(reorder, sizes, target)\n",
    "d2ltvm.plot_gflops(sizes, [np_gflops, default_gflops, reorder_gflops],\n",
    "            ['numpy', 'default', 'reorder'])\n"
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   "source": [
    "5.4. Parallelization¶\n",
    "Let’s revisit the pseudo code above. In the outermost for-loop for (x, 0, 64), each time we compute the results of a row in  C\n",
    " . Each row can be computed in parallel, so we can make the schedule parallelize axis x.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 64], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x: int32, 0, 64) \"parallel\" {\n",
      "    for (y.init: int32, 0, 64) {\n",
      "      C_3: Buffer(C_2, float32, [4096], [])[((x*64) + y.init)] = 0f32\n",
      "    }\n",
      "    for (k: int32, 0, 64) {\n",
      "      for (y: int32, 0, 64) {\n",
      "        let cse_var_2: int32 = (x*64)\n",
      "        let cse_var_1: int32 = (cse_var_2 + y)\n",
      "        C_3[cse_var_1] = (C_3[cse_var_1] + (A_3: Buffer(A_2, float32, [4096], [])[(cse_var_2 + k)]*B_3: Buffer(B_2, float32, [4096], [])[((k*64) + y)]))\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def parallel(n):\n",
    "    s, (A, B, C) = reorder(n)\n",
    "    s[C].parallel(C.op.axis[0])\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, args = parallel(64)\n",
    "print(tvm.lower(s, args, simple_mode=True))"
   ]
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  {
   "cell_type": "code",
   "execution_count": 47,
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     "metadata": {},
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   "source": [
    "parallel_gflops = bench_matmul_tvm(parallel, sizes, target)\n",
    "d2ltvm.plot_gflops(sizes, [np_gflops, default_gflops, reorder_gflops, parallel_gflops],\n",
    "            ['numpy', 'default', 'reorder', 'parallel'])"
   ]
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   "metadata": {},
   "source": [
    "6. Improve Cache Efficiency by Blocking¶\n",
    "In Section 5 we saw that properly reordering the loop axes to get more friendly memory access pattern, together with thread-level parallelization, could dramatically improve the performance for matrix multiplication. The results show that for small-scale matrices, our performance outperforms the NumPy baseline. However, for large matrices, we need to carefully consider the cache hierarchy discussed in Section 1.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import tvm\n",
    "from tvm import te\n",
    "import numpy as np\n",
    "import d2ltvm\n",
    "\n",
    "target = 'llvm -mcpu=skylake-avx512'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "sizes = 2**np.arange(5, 12, 1)\n",
    "exe_times = [d2ltvm.bench_workload(d2ltvm.np_matmul_timer(n)) for n in sizes]\n",
    "np_gflops = 2 * sizes **3 / 1e9 / np.array(exe_times)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "6.1. Blocked Matrix Multiplication¶\n",
    "\n",
    "![](https://tvm.d2l.ai/_images/matmul_block.svg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [64, 64], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [64, 64], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [64, 64], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x.outer.y.outer.fused: int32, 0, 4) \"parallel\" {\n",
      "    for (x.inner.init: int32, 0, 32) {\n",
      "      C_3: Buffer(C_2, float32, [4096], [])[ramp((((floordiv(x.outer.y.outer.fused, 2)*2048) + (x.inner.init*64)) + (floormod(x.outer.y.outer.fused, 2)*32)), 1, 32)] = broadcast(0f32, 32)\n",
      "    }\n",
      "    for (k.outer: int32, 0, 16) {\n",
      "      for (x.inner: int32, 0, 32) {\n",
      "        let cse_var_5: int32 = (floormod(x.outer.y.outer.fused, 2)*32)\n",
      "        let cse_var_4: int32 = ((floordiv(x.outer.y.outer.fused, 2)*2048) + (x.inner*64))\n",
      "        let cse_var_3: int32 = ((k.outer*256) + cse_var_5)\n",
      "        let cse_var_2: int32 = (cse_var_4 + (k.outer*4))\n",
      "        let cse_var_1: int32 = (cse_var_4 + cse_var_5)\n",
      "         {\n",
      "          C_3[ramp(cse_var_1, 1, 32)] = (C_3[ramp(cse_var_1, 1, 32)] + (broadcast(A_3: Buffer(A_2, float32, [4096], [])[cse_var_2], 32)*B_3: Buffer(B_2, float32, [4096], [])[ramp(cse_var_3, 1, 32)]))\n",
      "          C_3[ramp(cse_var_1, 1, 32)] = (C_3[ramp(cse_var_1, 1, 32)] + (broadcast(A_3[(cse_var_2 + 1)], 32)*B_3[ramp((cse_var_3 + 64), 1, 32)]))\n",
      "          C_3[ramp(cse_var_1, 1, 32)] = (C_3[ramp(cse_var_1, 1, 32)] + (broadcast(A_3[(cse_var_2 + 2)], 32)*B_3[ramp((cse_var_3 + 128), 1, 32)]))\n",
      "          C_3[ramp(cse_var_1, 1, 32)] = (C_3[ramp(cse_var_1, 1, 32)] + (broadcast(A_3[(cse_var_2 + 3)], 32)*B_3[ramp((cse_var_3 + 192), 1, 32)]))\n",
      "        }\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "tx, ty, tk = 32, 32, 4  # tile sizes\n",
    "\n",
    "def block(n):\n",
    "    A, B, C = d2ltvm.matmul(n, n, n)\n",
    "    s = te.create_schedule(C.op)\n",
    "    # Tile by blocks, and then parallelize the computation of each block\n",
    "    xo, yo, xi, yi = s[C].tile(*C.op.axis, tx, ty)\n",
    "    xy = s[C].fuse(xo, yo)\n",
    "    s[C].parallel(xy)\n",
    "    # Optimize the computation of each block\n",
    "    ko, ki = s[C].split(s[C].op.reduce_axis[0], factor=tk)\n",
    "    s[C].reorder(ko, xi, ki, yi)\n",
    "    s[C].vectorize(yi)\n",
    "    s[C].unroll(ki)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, (A, B, C) = block(64)\n",
    "print(tvm.lower(s, [A, B, C], simple_mode=True))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 450x300 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import d2ltvm\n",
    "blocked_gflops = d2ltvm.bench_matmul_tvm(block, sizes, target)\n",
    "d2ltvm.plot_gflops(sizes, [np_gflops, blocked_gflops], ['numpy', 'block'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "6.2. Write Cache\n",
    "\n",
    "The non-continuous write issue is severer than the non-continuous read. This is because we read once of each submatrix of A and B, but need to write by n times for the submatrix of C. In the following code block, we first write the results into a local buffer for each submatrix computation, and then write them back to C. It can be done by the cache_write method. We specify the buffer being used for each block by placing it within the yo axis using the compute_at primitive. The rest optimization is the same as before, but note that we need to use s[Cached] instead of s[C] to optimize the submatrix computation.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@main = primfn(A_1: handle, B_1: handle, C_1: handle) -> ()\n",
      "  attr = {\"from_legacy_te_schedule\": True, \"global_symbol\": \"main\", \"tir.noalias\": True}\n",
      "  buffers = {A: Buffer(A_2: Pointer(float32), float32, [512, 512], []),\n",
      "             B: Buffer(B_2: Pointer(float32), float32, [512, 512], []),\n",
      "             C: Buffer(C_2: Pointer(float32), float32, [512, 512], [])}\n",
      "  buffer_map = {A_1: A, B_1: B, C_1: C} {\n",
      "  for (x.outer.y.outer.fused: int32, 0, 256) \"parallel\" {\n",
      "    allocate(C.local: Pointer(local float32), float32, [1024]), storage_scope = local {\n",
      "      for (x.c.init: int32, 0, 32) {\n",
      "        C.local_1: Buffer(C.local, float32, [1024], [], scope=\"local\")[ramp((x.c.init*32), 1, 32)] = broadcast(0f32, 32)\n",
      "      }\n",
      "      for (k.outer: int32, 0, 128) {\n",
      "        for (x.c: int32, 0, 32) {\n",
      "          let cse_var_3: int32 = (x.c*32)\n",
      "          let cse_var_2: int32 = ((k.outer*2048) + (floormod(x.outer.y.outer.fused, 16)*32))\n",
      "          let cse_var_1: int32 = (((floordiv(x.outer.y.outer.fused, 16)*16384) + (x.c*512)) + (k.outer*4))\n",
      "           {\n",
      "            C.local_1[ramp(cse_var_3, 1, 32)] = (C.local_1[ramp(cse_var_3, 1, 32)] + (broadcast(A_3: Buffer(A_2, float32, [262144], [])[cse_var_1], 32)*B_3: Buffer(B_2, float32, [262144], [])[ramp(cse_var_2, 1, 32)]))\n",
      "            C.local_1[ramp(cse_var_3, 1, 32)] = (C.local_1[ramp(cse_var_3, 1, 32)] + (broadcast(A_3[(cse_var_1 + 1)], 32)*B_3[ramp((cse_var_2 + 512), 1, 32)]))\n",
      "            C.local_1[ramp(cse_var_3, 1, 32)] = (C.local_1[ramp(cse_var_3, 1, 32)] + (broadcast(A_3[(cse_var_1 + 2)], 32)*B_3[ramp((cse_var_2 + 1024), 1, 32)]))\n",
      "            C.local_1[ramp(cse_var_3, 1, 32)] = (C.local_1[ramp(cse_var_3, 1, 32)] + (broadcast(A_3[(cse_var_1 + 3)], 32)*B_3[ramp((cse_var_2 + 1536), 1, 32)]))\n",
      "          }\n",
      "        }\n",
      "      }\n",
      "      for (x.inner: int32, 0, 32) {\n",
      "        for (y.inner: int32, 0, 32) {\n",
      "          C_3: Buffer(C_2, float32, [262144], [])[((((floordiv(x.outer.y.outer.fused, 16)*16384) + (x.inner*512)) + (floormod(x.outer.y.outer.fused, 16)*32)) + y.inner)] = C.local_1[((x.inner*32) + y.inner)]\n",
      "        }\n",
      "      }\n",
      "    }\n",
      "  }\n",
      "}\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "def cached_block(n):\n",
    "    A, B, C = d2ltvm.matmul(n, n, n)\n",
    "    s = te.create_schedule(C.op)\n",
    "    # Create a write cache for C\n",
    "    CachedC = s.cache_write(C, 'local')\n",
    "    # Same as before, first tile by blocks, and then parallelize the\n",
    "    # computation of each block\n",
    "    xo, yo, xi, yi = s[C].tile(*C.op.axis, tx, ty)\n",
    "    xy = s[C].fuse(xo, yo)\n",
    "    s[C].parallel(xy)\n",
    "    # Use the write cache for the output of the xy axis, namely a block.\n",
    "    s[CachedC].compute_at(s[C], xy)\n",
    "    # Same as before to optimize the computation of a block .\n",
    "    xc, yc = s[CachedC].op.axis\n",
    "    ko, ki = s[CachedC].split(CachedC.op.reduce_axis[0], factor=tk)\n",
    "    s[CachedC].reorder(ko, xc, ki, yc)\n",
    "    s[CachedC].unroll(ki)\n",
    "    s[CachedC].vectorize(yc)\n",
    "    return s, (A, B, C)\n",
    "\n",
    "s, (A, B, C) = cached_block(512)\n",
    "print(tvm.lower(s, [A, B, C], simple_mode=True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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